The second example is given by quasi-geostrophic shallow-water equations with parameter \(\lambda >0\) , shorten in what follows into \((QGSW)_{\lambda }\) , which is a geophysical asymptotic model describing the circulation of the atmosphere at large time and space scales [1]}. This model is given by
\(\partial _tq^{\textnormal {\tiny {SW}}}+\mathbf {v}_{\lambda }^{\textnormal {\tiny {SW}}}\cdot \nabla q^{\textnormal {\tiny {SW}}}=0,\qquad \mathbf {v}_{\lambda }^{\textnormal {\tiny {SW}}}=\nabla ^{\perp }\Psi _{\lambda }^{\textnormal {\tiny {SW}}},\qquad (\Delta -\lambda ^2)\Psi _{\lambda }^{\textnormal {\tiny {SW}}}=q^{\textnormal {\tiny {SW}}}.\) 
